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Central Limit Theorem

     The central limit theorem (CLT) is one of the most useful theorems in statistics. It states that regardless of the population distribution, as long as all the variables are independently and identically distributed (i.i.d.), the sampling distribution of means will approach a normal distribution with mean equal to the population mean as the sampling size increases.

     Consider the population of Canada and imagine that you would like to find out what the mean height is. You start by taking a sample of size N and take the mean of that sample. But, this does not guarantee at all, that this will be the same as the population's mean. Now imagine that you continue to take more random samples of size N and find their means. Plotting all those means will give you a sampling distribution of means. The CLT states that this sampling distribution will approach a normal distribution as N grows large. Furthermore the mean of this sampling distribution will be approximately equal to the population mean as well. The sampling distribution variance will be approximately the population variance divided by N. The sampling standard deviation will be the square root of that. The central limit theorem evidently gives the ability to solve for the population mean with more accuracy than using one sample.  

Normal Distribution on a Portfolio

1) The daily returns on a portfolio are normally distributed with a mean of 0.001 and a standard deviation of 0.002. What is the probability that the average return for the portfolio over the next 100 days exceeds 0.0015? 2) In May 1983, after an extensive investigation by the Consumer Product Safety Commission, Honeywell

Normal Approximation to Binomial Distribution - Weld Example

Smith is a weld inspector at a ship yard. He knows from keeping track of good and substandard welds that 5% will be substandard. If he checks 300 of 7500 welds, what is the probability that he will find less than 20 substandard welds? To solve the problems on the up-coming test we will use either the "normalcdf" or InvNorm" f

Statistical Analysis: Are we smarter than our parents

Read the article entitled, "Are We Smarter than Our Parents?" in chapter 5 of your textbook. This article addresses a study by Dr. James Flynn of the rise of the IQ rate over generations, and how statistics are involved in tracking this phenomenon, especially with reference to the material in chapter 5. Write a paper in which

Central Limit Theorem and Probabilities Under Standard Normal Curve

Will you check my work on 1, 2 & 5 and then help me with 3, 4 & 6... I don't know if the formula is still the same for all of them or different. 1. The heights of eighteen-year-old men are approximately normally distributed with a mean (µ) of 68 inches and a standard deviation (σ) of 3 inches. What is the probability that

Central Limit Theorem Explained in Layman's Terms

Could you please explain Central Limit Theorem to me in layman's terms and then read the article linked below and detail how it was used therein? Thank you.

The solution gives detailed steps on determining type I and type II errors in a specific hypothesis testing and using confidence interval method to make the conclusion of the same test. Next, central limit theorem is well defined and explained in details.

(b) The quality control engineer for a furniture manufacturer is interested in the mean amount of force necessary to produce cracks in stressed oak furniture. She performs a two-tailed test of the null hypothesis that the mean for the stressed oak furniture is 650. What are Type I and Type II errors for this problem? (c) A 95

Application of Probability in Duracell Battery Life

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Insurance Agents Nationwide in Ohio

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Statistics Control Limits

A tire company is interested in monitoring the process that produced tread thickness on its tires. Every hour 4 tires are selected from production and the tread thickness is measured. Data for the past 25 days is shown as follows: (Please see the attached file.) a. What type of control chart would you recommend be used i

Center Line and Control Limits

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Central Limit Theorem and the Poisson Distribution

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Normal Distribution and Central Limit Theorem

The weight of 3rd grade children is normally distributed with a mean of 64 and a standard deviation of 10. Two children are selected at random from a very large population. (a.) What is the probability that both will weigh more than 70 pounds? (b.) What is the probability that one of the children will weigh more than 65 poun

Central limit theorem..

The average balance for customer accounts in The Reserve Fund at the time it was frozen by the Securities Exchange Commission (SEC) was $22,500, with a standard deviation of $7,500. The SEC overseers want to draw a sample of 100 accounts to help assess the impact of the fund's freeze on the account holders. Precisely (that is, u

Central Limit Theorem Conversation

Please see attachment. And please provide clear and thorough information so that I can understand it in future problems. Please provide assistance in the following 10 parts: 1. Boss has a question about a population: 2. To costly or can't do population so forced to sample 3. With sampling, will be sampling error (CLT


In the Department of Education at UR University, student records suggest that the population of students spends an average of 5.5 hrs per week playing organized sports. The population's standard deviation is 2.2 hrs per week. Based on a sample of 121 students, healthy lifestyles Inc. (HLI) would like to apply the central limit t

Normal Approximation to Binomial and Central Limit Theorem

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See attached file and please solve calculations in red. 2 Using the descriptive statistics data determined during Week One's weekly problem discussion, the mean for EI followed a standard distribution with a mean of 132.83 and a standard deviation of 15.68. If we selected another random sample of 50 participant

Statistics: 15 Problems

1. A population is normally distributed, with a mean of 23.45 and a standard deviation of 3.8. What is the probability of each of the following? a. Taking a sample of size 10 and obtaining a sample mean of 22 or more b. Taking a sample of size 4 and getting a sample mean of more than 26 2. The Statistical Abstract of the

Statistics: Normal Distribution, probabilities, values, central limit theorem

Included with each section or problem are reference examples and end of section exercises that can be used as a guide. Be sure to show your work in case partial credit is awarded. To receive full credit, work must be shown if applicable. Section 5.1: Introduction to Normal Distribution and the Standard Normal Distribution 1.

Statistics: 8.46, 8.62, Central Limit Theorem, Population shape of concern

8.46 A random sample of 10 miniature Tootsie Rolls was taken from a bag. Each piece was weighed on a very accurate scale. The results in grams were 3.087 3.131 3.241 3.241 3.270 3.353 3.400 3.411 3.437 3.477 (a) Construct a 90 percent confidence interval for the true mean weight. (b) What sample size would be necessary

Central limit

A worldwide organization of academics claims that the mean IQ score of its members is 118, with a standard deviation of 15. A randomly selected group of 40 members of this organization is tested, and the results reveal that the mean IQ score in this sample is 116.8. If the organization's claim is correct, what is the probability

Central Limit Theorem and Population Shape

1. State the main points of the Central Limit Theorem for a mean. 2. Why is population shape of concern when estimating a mean? What does sample size have to do with all?

Distribution characteristics foundations

The normal distribution has several characteristics that make it the underlying foundation of much of inferential statistical tools. List these characteristics and explain how they contribute to the power of the distribution as foundation of inferential decision making.

Statistics discussion: Control chart distributions differences for a large sample

Please help with the following problem. Provide at least 200 words in the solution. Control charts are a popular (and often reasonable) way to statistically monitor a processes' performance. They are used for both variables (i.e. quantitative measures) and attributes (qualitative measures). Our authors state that the underly

Central limit theorem for population

Sample sizes of 50 are selected from a normal population with a mean of 78.1 and standard deviation of 16.2. Calculate the probability that a single value selected at random will be between 71.1 and 85.1

The Central Limit Theorem

The Central Limit Theorem The Sony Corporation produces a Walkman that requires two AA batteries. The mean life of these batteries in this product is 35.0 hours. The distribution of the battery lives closely follows the normal probability distribution with a standard deviation of 5.5 hours. As a part of their testing program

Sampling Distribution/Central Limit Theorem

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